Self-similarities of periodic structures for a discrete model of a two-gene system

We report self-similar properties of periodic structures remarkably organized in the two-parameter space for a two-gene system, described by two-dimensional symmetric map. The map consists of difference equations derived from the chemical reactions for gene expression and regulation. We characterize the system by using Lyapunov exponents and isoperiodic diagrams identifying periodic windows, denominated Arnold tongues and shrimp-shaped structures. Period-adding sequences are observed for both periodic windows. We also identify Fibonacci-type series and Golden ratio for Arnold tongues, and period multiple-of-three windows for shrimps. (C) 2012 Elsevier B.V. All rights reserved.